I like this question a lot and I think that it's an important one. So here goes a (necessarily incomplete) attempt at answering such a broad and personal question.
First, "motivation" and "understanding for the essence" can mean very different things. There is of course physical motivation and intuition, and that probably applies most immediately to the Calculus III course that you are talking about. E.g. for the concept of derivatives of vector valued functions, you can think of the vector valued function of time that gives the position of an object as a vector. Of course, its derivative with respect to time will be the velocity (also a vector, since it described the speed and the direction of the movement) and the second derivative will be the acceleration. A good course in such an applicable subject will not just ask question like "compute the derivative of such and such a function", but will actually confront the student with real life examples.
But there is also intuition for less physical and more platonic concepts, such as that of a group, or of a prime number. Again, examples help. Also, you should always try to ask yourself the question "Could I have invented this?". If you see a new definition, ask yourself "What concrete problem might have prompted someone to define such a thing?". If you see a new result, ask yourself "Why was this to be expected, why would it be at least a reasonable conjecture?". Then try to convert your intuition into a proof. When you see a proof, ask yourself "Why is this a natural approach to try? Could I have proven this?". I agree with you that knowing the historical development can be very helpful in this and you should invest time in researching it.
I would like to contradict you in your assertion that intuition, motivation and historical context are black magic secrets that mathematicians acquire and then keep to themselves. It is true of some books and some teachers. So, you just have to find the right books. For that, you could ask for a specific recommendation here, including the area you want to learn and the books you have looked at, together with the reason you found them deficient. Of course, you can also ask specific "intuition" type questions.
To learn to appreciate mathematics, it is important to think about mathematics in your "spare time". Go out into nature and think about what your lecturer just told you in the last lecture. Or just think about whatever you find interesting. Then come back home with specific questions and look them up or ask them here.
Finally, something that I preach my students all the time is that they should develop a critical approach to what they are taught: if I give them a definition, they should try to come up with as many examples as possible. If a state a theorem of the type "A implies B", they should go home and find an example that "B does not necessarily imply A". If they do find such an example, they should ask themselves what additional hypotheses they need to impose to get the converse. If they don't, they should come back to me and ask me "but you haven't told us the whole story. What about the converse?".
In short, don't expect your lecturers to tell you everything you need to know. You should expect to have to think, to investigate yourself, to ask questions, and, above all, to think about mathematics because you can't help it, rather than because you are told to. This is not something, most people are born with, it's something that you have to cultivate.
Your notation looks fine. You could also use the more informal $\alpha = \max(\{f(x_1),\ldots,f(x_n)\})$ or even $\alpha = \max(f(x_1),\ldots,f(x_n))$.
Finally, you could say that $\alpha$ is the maximum (or maximal) value among $f(x_1),\ldots,f(x_n)$, or that $\alpha$ is the maximum (or maximal) value attained by $f$ on the points $x_1,\ldots,x_n$.
Best Answer
I believe the terms meet and join come from lattice theory. A lattice, after all, can be defined as a partially ordered set in which any two elements have a meet and a join. In practice, a lattice typically arises as a collection of "closed" sets (with respect to some kind of algebraic closure) ordered by set inclusion; typical examples would be the lattice of all subgroups of a group, or the lattice of all subspaces of a vector space.
Consider the lattice of subspaces of a vector space. The meet of two subspaces is their set-theoretic intersection; e.g., for two $2$-dimensional subspaces of $\mathbb R^3$, their meet is the line where the two planes meet. The join of two subspaces is what we get when the two subspaces join together to make a bigger subspace; in general it's not just the set-theoretic union, but the linear span of the union.
You also wanted to know where the symbols $\vee$ and $\wedge$ come from. I don't know but I'd guess they are derived from the symbols $\cup$ and $\cap$ for union and intersection, the lattice operations in the lattice of all subsets of a set. As for the symbols $\cup$ and $\cap$, my wild guess is that they are stylized versions of the letters u (for union) and n (for intersection). And if that's not the true history, it's good enough for a mnemonic, isn't it?